Questions tagged [stable-homotopy-category]
The stable-homotopy-category tag has no usage guidance.
68
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Definition of an E-infinity algebra
Can anyone give me a plain-and-simple
definition of an E-infinity algebra without using
the words "operad," "ring spectrum," or
"stable homotopy"?
Sorry, but I honestly couldn't find it using
all on-...
- 3,257
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votes
2
answers
3k
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Why the sphere spectrum is more correct than $\mathbb{Z}$?
One may argue that $\mathbb{S}$ is more correct than $\mathbb{Z}$. Can anyone make it more explicitly? For example, what information will be lost if we work in $\mathbb{Z}$ instead of $\mathbb{S}$？
...
- 1,084
21
votes
4
answers
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Multiplicative Structures on Moore Spectra
The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
- 2,277
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3
answers
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Is there an additive model of the stable homotopy category?
$\DeclareMathOperator\Ho{Ho}$Is there a model category $C$ on an additive category such that its homotopy category $\Ho(C)$ is the stable homotopy category of spectra and the additive structure on $\...
- 1,315
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2
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Difficulties with the mod 2 Moore Spectrum
I have been informed that there is a reference out there which specifically details what goes wrong with the mod 2 Moore spectrum, i.e. why it is not $A_\infty$ or something? I do not know the details,...
- 10.1k
14
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1
answer
651
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localizing subcategories of $HF_p$-local spectra
This entire question takes place in the $HF_p$-local category of $p$-local spectra, i.e. the essential image of $HF_p$-localization on the stable homotopy category. $HF_p$ itself is in there, and of ...
- 489
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2
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Is there a constructive description of type in the p-local stable homotopy category?
The title pretty much sums it up - but let me give a little bit of background first.
In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) ...
- 8,561
13
votes
4
answers
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Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there).
The answer ...
13
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1
answer
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Equivalent definitions of Thom spectra
Background and notations:
Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...
- 552
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2
answers
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Connective spectra and infinite loop spaces
It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective.
For me, an infinite loop ...
- 9,591
11
votes
2
answers
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Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$?
I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably ...
- 16.5k
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0
answers
383
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How does the HHR Norm functor interact with the cotensor over $G$-spaces?
Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces $...
- 423
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0
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Fields in Stable Homotopy Theory
It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...
- 10.1k
10
votes
1
answer
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Morava $K(n)$'s are not $E_{\infty}$
I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right.
...
- 2,277
10
votes
1
answer
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Smashing localizations in the category of spectra
Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization.
The functor $L_E$ generally does not ...
- 25.3k
10
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1
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When is the Thom spectrum of a virtual vector bundle effective?
Remark: My question is valid in the classic setting of the stable homotopy category of spectra of CW-complexes. An answer on that setting will also be valid.
Denote as $SH(X)$ Voevodsky's stable ...
- 2,741
8
votes
2
answers
995
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Why do we study complex orientable cohomology theories
It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology ...
- 2,035
8
votes
1
answer
478
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Stabilization of $\infty$-categories versus SW stabilization
Spanier-Whitehead stabilization provides a way to extend a category $\bf E$ to a bigger one $\mathcal{SW}_\Omega(\bf E)$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}_\...
- 13.2k
8
votes
1
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Higher coherent multiplicative structures on S-algebras
In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra ...
- 10.1k
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1
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488
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Inverting objects in a symmetric monoidal category
In Voevodsky’s ICM address:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/voevodsky-a1-homotopy-theory-icm-1998.pdf
In theorem 4.3 it is claimed that given a symmetric monoidal ...
- 103
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1
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551
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Is there an obvious reason why p-localization of spectra is a finite localization?
Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the ...
- 1,841
8
votes
1
answer
216
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Bousfield's distributive lattice DL and non-ring spectra
Bousfield, in his paper "The Boolean algebra of spectra" (Comm Math Helv 54, 368–377 (1979), https://doi.org/10.1007/BF02566281), defined $\mathbf{DL}$, a sublattice of the Bousfield lattice,...
- 4,146
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votes
1
answer
404
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Is $[X, \_]$ a homology theory?
Let $X$ be a CW-spectrum. It is well-known that $[\_ ,X]$ is a generalized cohomology theory and, by Brown's representability theorem, every generalized theory is $H$ represented by a spectrum (namely,...
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2
answers
516
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Chromatic convergence of E(n)-localized homotopy categories
Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy ...
- 10.1k
7
votes
2
answers
417
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Model category structure on spectra
I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.
Let $S$ be a finite dimensional Noetherian scheme ...
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votes
1
answer
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Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum
I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra).
Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$,...
- 5,492
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0
answers
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Homotopy theory of differential objects
In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
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0
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Self-equivalences of the stable homotopy category
I have recently started approaching Stable Homotopy Theory and came up with what is probably a rather naive question to ask, though it looks like I can not find references on it around.
Let $\mathcal{...
- 201
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0
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Is there a list of examples of orthogonal spectra?
Schwede's symmetric spectra book project provides point-set models of many important spectra as symmetric spectra, including (in §I.1) the sphere spectrum, Eilenberg-Mac Lane spectra, several Thom ...
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2
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Does a finite suspension spectrum make a space finite?
Suppose that $X$ is a space whose suspension spectrum $\Sigma_+^\infty(X)$ is dualizable in the stable homotopy category. I believe this is equivalent to saying that $\Sigma_+^\infty(X)$ is (weakly) ...
- 65.1k
6
votes
2
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518
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Coreflective Subcategories of the Stable Homotopy Category
Here by stable homotopy category I mean the homotopy category of spectra, or maybe just some monogenic, Brown, algebraic, etc. stable homotopy category (in the language of Hovey, Palmieri and ...
- 10.1k
6
votes
1
answer
251
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Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum
We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we ...
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2
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Smash product of spheres in $\mathbf{SH}$ and product in cohomology
I have two very concrete and simple question. Just in case I write downwards what led me into this.
My questions: Let $\mathbf{SH}(X)$ be the stable homotopy category of Voevodsky. Denote $S^n$ the ...
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6
votes
1
answer
444
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Stable Adams operations
I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
6
votes
1
answer
381
views
When was the word "stable" first used to describe stable homotopy theory?
The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things:
Homotopy groups stabilize after taking suspensions (...
- 3,301
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votes
0
answers
217
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Compatible algebraic Spanier-Whitehead dual
Let me first ask an intuitive version of the question:
Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we ...
- 2,013
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votes
1
answer
572
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Is the stable homotopy category idempotent complete?
Is the stable homotopy category idempotent complete? I have not been able to prove it, and the proof for abelian groups seems to strongly rely on looking at elements.
Thanks,
Jon
- 10.1k
5
votes
0
answers
122
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Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$
My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$.
It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...
- 2,741
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0
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Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object
Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield ...
- 10.1k
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votes
1
answer
472
views
Does the (singular)cohomology of any acyclic spectrum vanish?
I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or $H\mathbb{...
- 16.5k
4
votes
0
answers
61
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Endomorphism in the rational stable $O(2)$-equivariant category of the universal space of the family of finite dihedral subgroups
Let $G$ be a compact Lie group. We can define $\mathfrak{F}G$ to be the collection of conjugacy classes of closed subgroups of $G$ whose Weyl group is finite, a bi-invariant metric on $G$ induces a ...
- 757
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votes
0
answers
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Direct image and infinite suspension
I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
- 2,741
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votes
0
answers
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Unstable and stable looping and delooping
I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an ...
- 2,741
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votes
0
answers
368
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matrix ring spectra
I am trying to understand matrix ring spectra. Apparently, I have two different definitions of those and I did not manage to show that they are equivalent - maybe they even are not in the general case....
- 7,537
3
votes
3
answers
490
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Bousfield Classes
This question has a few parts:
1) Is the Bousfield class of $\langle E\rangle$ the class of $E$-acyclics, i.e. $\langle E\rangle=\left\{ X:E\wedge X=0\right\}$ or is it the class of spectra which are ...
- 10.1k
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votes
2
answers
284
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Is the "inverse" (i.e., the "cohomological") numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra "acceptable"? [closed]
I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider ...
- 16.5k
3
votes
1
answer
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Basic questions on spectra
I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme.
Let $E$ be an $\Omega$-spectrum and $\varphi \colon ...
- 2,741
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votes
1
answer
158
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Basic question on the cobordism spectrum
I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and ...
- 2,741
3
votes
1
answer
286
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Infinite loop space of ring spectra: the cup product
I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory.
Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
- 2,741
3
votes
1
answer
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Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension
I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...
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